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developed. Eisenstein insisted on beautiful formulae, somewhat
in the manner of a modernized Euler; Riemann wanted to intro-
duce the complex variable and derive the entire theory, with a
minimum of calculation, from a few simple, general principles.
Thus, no doubt, originated at least the germs of one of Rie-
mann's greatest contributions to pure mathematics. As the
origin of Riemann's work in the theory of functions of a com-
plex variable is of considerable importance in his own history
and in that of modern mathematics, we shall glance at what is
known about it.
Briefly, nothing definite. The definition of an analytic func-
tion of a complex variable, discussed in connexion with Gauss'
anticipation of Cauchy's fundamental theorem, was essentially
that of Riemann. "When expressed analytically instead of geo-
metrically that definition leads to the pair of partial differential
equations* which Riemann took as his point of departure for a
theory of functions of a complex variable. According to Dede-
kind, 'Riemann recognized in these partial differential equa-
tions the essential definition of an [analytic] function of a com-
plex variable. Probably these ideas, of the highest importance
for his future career, were worked out by him hi the fall vaca-
tion of 1847 [Riemann was then twenty-one] for the first tune.'
Another version of the origin of Riemann's inspiration is due
to Sylvester, who tells the following story, which is interesting .
even if possibly untrue. In 1896, the year before his death,
Sylvester recalls staying at 'a hotel on the river at Nuremberg,
where I conversed outside with a Berlin bookseller, bound, like
myself, for Prague. ... He told me he was formerly a fellow
pupil of Riemann, at the University, and that, one day, after
receipt of some numbers of the Comptes rendus from Paris, the
latter shut himself up for some weeks, and when he returned to
* If 2 = x 4- iy, and to = u + iv, is an analytic function of 2,
Rieniann's equations are
1?  ^?    ?? __      3o
15 ~ djf    ty ~~ ~ Tx
These equations had been given much earlier by Cauchy, and even
Cauchy -was not the first, as D'Alembeit had stated the aquations in
the eighteenth century.